In a random experiment of throwing $5$ coins, the number of heads is defined as a random variable. The mean of the random variable is

Given,

In a random experiment of throwing $5$ coins, the number of heads is defined as a random variable, so making the table of probability function we get,

$\left[\begin{array}{c}X\\ P\left(X\right)\end{array}\begin{array}{cccccc}0& 1& 2& 3& 4& 5\\ \frac{1}{2^5}& \frac{5}{2^5}& \frac{10}{2^5}& \frac{10}{2^5}& \frac{5}{2^5}& \frac{1}{2^5}\end{array}\right]$

Now we know that the mean of the random variable is $\sum _{n=1}^5{X}_{n}P\left(X_n\right)$

So, $\sum _{n=1}^5{X}_{n}P\left(X_n\right)=0+\frac{5}{2^5}+\frac{20}{2^5}+\frac{30}{2^5}+\frac{20}{2^5}+\frac{5}{2^5}$

$\Rightarrow \sum _{n=1}^5{X}_{n}P\left(X_n\right)=\frac{80}{2^5}=\frac{80}{32}=\frac{5}{2}$